Magic identities for conformal four-point integrals

TL;DR

The paper investigates conformal four-point integrals in four-dimensional field theories, focusing on off-shell cases and the equivalence of the three-loop ladder (triple scalar box) and tennis court integrals. It introduces a simple turning-symmetry argument on a shared two-loop subintegral and an iterative slingshot rule to generate an infinite family of higher-loop equalities, with a differential-operator construction to address contact terms. An independent Mellin–Barnes analysis provides explicit integral representations and confirms the identity by showing a variable transformation relating the triple box and tennis court, supporting Phi^{(3)} = Psi^{(3)}. The work offers a general framework for constructing and validating conformal four-point integral identities, with potential implications for perturbative structures and integrability in N=4 SYM.

Abstract

We propose an iterative procedure for constructing classes of off-shell four-point conformal integrals which are identical. The proof of the identity is based on the conformal properties of a subintegral common for the whole class. The simplest example are the so-called `triple scalar box' and `tennis court' integrals. In this case we also give an independent proof using the method of Mellin--Barnes representation which can be applied in a similar way for general off-shell Feynman integrals.

Figures (12)

• Figure 1: The one-loop ladder integral. Each line represents a propagator with the integration point given by a solid vertex. The reason for the names ladder and box is clearer in the momentum representation of the same integral.
• Figure 2: The two-loop ladder integral. The dashed line represents the numerator $x_{24}^2$.
• Figure 3: The two-loop turning identity obtained from the pairwise point swap, $x_1 \longleftrightarrow x_2$, $x_3 \longleftrightarrow x_4$.
• Figure 4: Two examples of three-loop conformal four-point integrals, the three-loop ladder and the 'tennis-court'.
• Figure 5: The three-loop ladder expressed as the integral of the two-loop ladder against the 'slingshot'. The empty vertex is the point $x_5$ which must be identified with the point $x_5$ from the two-loop ladder sub-integral before being integrated over.